On some questions concerning a functional equation involving Möbius transformations

Given a field double-struck F sign, is it true that any bijection which preserves the single operation (x, y) → (x + y)/(x - y) is necessarily a field automorphism? We show that the answer is "yes" for double-struck F sign = ℚ, ℝ, double-struck F signp with p ≠ 5, or if double-struck F sign is a Galois extension of ℚ of degree 2k, and "no" for double-struck F sign = double-struck F sign5. © Birkhäuser Verlag, Basel, 2000.